The present invention relates to the control of the propagation of elastic waves such as vibrations, notably in the fields of mechanics and of geology. In particular, the present invention relates to the bending of transversal elastic waves propagating in a thin plate.
Significant progress has recently been made in the control of electromagnetic waves. Transformations based upon Maxwell's equations in a cylindrical coordinate system allow structures to be defined for bending electromagnetic waves around a region to hide. Such a structure, which may also be termed an “invisibility cloak”, is a metamaterial having a permeability and a permittivity that are strongly heterogeneous and anisotropic and allow the electromagnetic waves to bend around the region to hide. The term metamaterial here designates an artificial composite material having physical properties that are not found in a natural material. The permeability and the permittivity of the metamaterial may be deduced from a transformation of a coordinate system of Maxwell's equations.
United States Patent Application Publication No. 2008/0024792 discloses a geometric transformation allowing for the definition of an invisibility cloak with respect to light. This geometric transformation leads to permittivity and permeability tensors, anisotropic and varying in space, that may be approximated with the aid of a periodic structure comprising structural elements such as “split ring resonators” or in the form of a “Swiss Roll”. The invisibility properties of this structure are based on the resonance of the structural elements, and therefore act intrinsically in a specific frequency range.
United States Patent Application Publication No. 2008/0165442 uses the same geometric transformation as that of the application US 2008/0024792, but proposes to approximate the permittivity and permeability tensors with the aid of other structural elements having a fixed permeability, and presenting the shape of metallic lengthened ellipsoids.
United States Patent Application Publication No. 2009/0218523 proposes the use of gradient index materials to approximate the permittivity and permeability tensors.
Contrary to Maxwell's equations and as described in the previously-mentioned documents, Navier equations describe the propagation of elastic waves that do not remain invariable with respect to geometric transformations of the coordinate system. It turns out that such geometric transformations are not applicable to Navier equations. Nevertheless, in a cylindrical coordinate system, equations relating to waves transversal to their propagation plane appear to be unrelated to equations concerning longitudinal and shear waves situated in the propagation plane to which they remain associated. Document [1] “Achieving control of in-plane elastic waves”, M. Brun, S. Guenneau, and A. B. Movchan, Applied Physics Letters 94, 061903 (2009), describes a cylindrical structure adapted to elastic waves situated in their propagation plane. The propagation of these waves is described by a 4th rank (non-symmetric) elasticity tensor with 24 Cartesian inputs and an isotropic density. This document shows that the required properties of a metamaterial for bending elastic waves around a cylindrical zone require the intervention of a 4th rank elastic tensor and 34 Cartesian inputs variable in space. Nevertheless, in the particular case of a thin plate, that is to say having a large length and width with respect to its thickness, the elastic tensor may be represented in a cylindrical coordinate system by a diagonal matrix with two inputs variable in space.
Document [2] “Ultrabroadband Elastic Cloaking in Thin Plates”, M. Farhat, S. Guenneau, S. Enoch, Physical Review Letters, PRL 103, 024301(2009) describes an anisotropic heterogeneous structure for bending transversal elastic waves around a zone to protect of a thin plate. This structure is formed by a plurality of radially symmetric layers, each having a Young's modulus and a constant mass density. To determine the behavior of this structure in relation to elastic waves to be controlled, the wavelength of the elastic waves was considered to be very large with respect to the thickness of the plate and small with respect to the other dimensions of the plate, which allows the von Karman Theory hypotheses to be adopted (“Theory of plates and shells”, S. Timoshenko, McGraw-Hill, New York, 1940, and “Wave motion in elastic solids”, K. F. Graff, Dover, N.Y., 1975).
In a cylindrical coordinate system, a displacement u(0, 0, U(r,θ)) of the plate in a direction x3 perpendicular to the plane of the plate is a solution of the following differential equation:λ∇·{ζ−1∇└λ∇·(ζ−1∇U)┘}−β04U=0  (1)
in a zone protected by the annular structure formed in the plate, centered at the coordinate origin. In equation (1):
λ=ρ1/2(r), ρ being the mass density of the annular structure,
ζ is equal to E−1/2, E being a Young's modulus of the material of the plate,
∇ is the nabla operator in cylindrical coordinate
      (                  ∂                  /                      ∂            r                              ,                        1          r                ⁢                  ∂                      /                          ∂              θ                                            )    ,  and
β04=ω2ρ0h/D0, ω being the pulsation of elastic waves, ρ0 being the mass density of the material constituting the plate, h being the thickness of the plate, and D0 being the flexural rigidity of the plate.
The following coordinate transformation is then applied:r′=a+r(1−a/b)  (2)
wherein a and b are the interior and exterior radii of the annular structure centered on the coordinate origin. This transformation allows for compression of the region such that r<a in the ring (a<r<b). It results that by choosing a plate having a constant mass density, for example ρ0=1, the Young's modulus and the mass density components of the structure have the following values:
                                          E            r                    =                                                    (                                  b                                      b                    -                    a                                                  )                            4                        ⁢                                          (                                                      r                    -                    a                                    r                                )                            4                                      ,                                  ⁢                              E            θ                    =                                                                      (                                      b                                          b                      -                      a                                                        )                                4                            ⁢                                                          ⁢              and              ⁢                                                          ⁢              ρ                        =                                                            (                                      b                                          b                      -                      a                                                        )                                4                            ⁢                                                (                                                            r                      -                      a                                        r                                    )                                2                                                                        (        3        )            
r being comprised between a and b.
The annular structure thus presents an anisotropic Young's modulus E and an isotropic mass density ρ, E and ρ varying as functions of the radius only.
In document [2], the ideal structure defined by equations (3) is approximated by a structure formed by several concentric annular layers having Young's moduli respectively increasing from the interior layer towards the exterior layer. Nevertheless, a structure formed of several concentric annular layers having different Young's moduli is rather difficult to implement, since to get as close as possible to the ideal structure, the number of layers must be as high as possible.
It is therefore desired to define a structure for bending the transversal elastic waves propagating in a thin plate that is easy to fabricate.